Abstract
We investigate the upper tail probabilities of the all-time maximum of a stable Levy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Levy process. We also study the lower tail probabilities of the integrated stable Levy process in the presence of a power positive drift.
Highlights
Introduction and statement of the resultsLet L be a real strictly α-stable Lévy process with characteristic exponent Ψ(λ) = log(E[eiλL1 ]) = −αe−iπαρ sgn(λ) = − |λ|αeiπα(1/2−ρ) sgn(λ), λ ∈ R, where α ∈
It is well-known from e.g. Proposition 48.10 in [19] that
In the last part of the paper, we study the lower tail problem for the integrated stable process with a power positive drift
Summary
(b) Taking f (u) = 1{u≥rx} for some r > 0 and applying as above the compensation formula leads to the estimate This implies the following limit theorem for the law of the renormalized overshoot: Proposition 2.2. That the standard Pareto distribution with parameter δ > 0 has distribution function 1 − (r + 1)−δ on (0, ∞) This observation seems new even in the classical case of a linear drift γ = 1 with α > 1. By Remark 2 of [6], this implies that Kx converges at infinity to some proper random variable - see Theorem 4.2 in [13] for more general results. Let us refer to [10] for related results in the presence of a compound Poisson process
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